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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Computability.DegreeRigidity.Constructibility.HierarchyCertificatePower
import OAI.Computability.DegreeRigidity.Representation.AmbientDegreeSyntax
namespace OAI
namespace TuringRigidity.BoundedDefinability
open ElementaryModel BoundedSetTheory TransitiveNameModel SetModelFunctions RelativeConstructible
universe u
def DomainDefinable (M : ZFSet.{u})
(P : ZFSet.{u} → (ℕ → ZFSet.{u}) → Prop) : Prop :=
∃ p : SentenceForm, ∃ d : ℕ → ZFSet.{u}, ∃ k : ℕ,
(∀ i, ∃ j, j < k ∧ d i = d j) ∧ (∀ i, d i ∈ M) ∧
∀ N : ZFSet.{u}, Transitive N → (∀ i, d i ∈ N) →
∀ e, (∀ i, e i ∈ N) → (p.Sat (N : Set ZFSet) (mix e d) ↔ P N e)
variable {M : ZFSet.{u}} {P Q : ZFSet.{u} → (ℕ → ZFSet.{u}) → Prop}
theorem Definable.toDomain {P : (ℕ → ZFSet.{u}) → Prop} (h : Definable M P) :
DomainDefinable M (fun _ e => P e) := by
obtain ⟨p,d,hd,hp⟩ := h
let b := (fromBounded p).bound + 1
let d' : ℕ → ZFSet.{u} := fun i => d (if i < b then i else 0)
have hn (i : ℕ) : ∃ j, j < b ∧ d' i = d' j := by
by_cases hi : i < b
· exact ⟨i,hi,rfl⟩
· exact ⟨0,by dsimp [b]; omega,by simp [d',hi,show 0 < b by dsimp [b]; omega]⟩
have heq (e : ℕ → ZFSet.{u}) : p.Eval (mix e d') ↔ p.Eval (mix e d) := by
rw [← bounded_univ,← bounded_univ]
apply (fromBounded p).finite_support
intro i hi
unfold mix
split
· rfl
· dsimp [d']
rw [ite_eq_left (show i / 2 < b by dsimp [b]; omega)]
refine ⟨fromBounded p,d',b,hn,fun i => hd _,?_⟩
intro N hN hdN e he
rw [bounded_sat,Formula.absolute p N hN _ (by
intro i; unfold mix; split <;> first | exact he _ | exact hdN _)]
exact (heq e).trans (hp e)
theorem DomainDefinable.congr (h : DomainDefinable M P)
(heq : ∀ N e, P N e ↔ Q N e) : DomainDefinable M Q := by
obtain ⟨p,d,k,hk,hd,hp⟩ := h
exact ⟨p,d,k,hk,hd,fun N hN hdN e he => (hp N hN hdN e he).trans (heq N e)⟩
theorem DomainDefinable.and (hP : DomainDefinable M P) (hQ : DomainDefinable M Q) :
DomainDefinable M (fun N e => P N e ∧ Q N e) := by
obtain ⟨p,d,k,hk,hd,hp⟩ := hP
obtain ⟨q,c,l,hl,hc,hq⟩ := hQ
refine ⟨.conj (p.rename (slotMap id (fun i => 2*i)))
(q.rename (slotMap id (fun i => 2*i+1))),mix d c,2 * max k l,?_,?_,?_⟩
· intro i
by_cases hi : i % 2 = 0
· obtain ⟨j,hj,heq⟩ := hk (i/2)
refine ⟨2*j,by omega,?_⟩
rw [mix_even]
simpa only [mix,ite_eq_left hi] using heq
· obtain ⟨j,hj,heq⟩ := hl (i/2)
refine ⟨2*j+1,by omega,?_⟩
rw [mix_odd]
simpa only [mix,ite_eq_right hi] using heq
· intro i; unfold mix; split <;> first | exact hd _ | exact hc _
· intro N hN hdc e he
have hdN (i : ℕ) : d i ∈ N := by simpa only [mix_even] using hdc (2*i)
have hcN (i : ℕ) : c i ∈ N := by simpa only [mix_odd] using hdc (2*i+1)
have hd' : mix d c ∘ (fun i => 2*i) = d := by funext i; simp
have hc' : mix d c ∘ (fun i => 2*i+1) = c := by funext i; simp
change (p.rename _).Sat _ _ ∧ (q.rename _).Sat _ _ ↔ _
simp only [SentenceForm.sat_rename]
change p.Sat _ (mix e (mix d c) ∘ slotMap id (fun i => 2*i)) ∧
q.Sat _ (mix e (mix d c) ∘ slotMap id (fun i => 2*i+1)) ↔ _
rw [mix_slotMap,mix_slotMap,Function.comp_id,hd',hc',hp N hN hdN e he,hq N hN hcN e he]
theorem DomainDefinable.existsSet (h : DomainDefinable M P) :
DomainDefinable M (fun N e => ∃ x ∈ N, P N (cons x e)) := by
obtain ⟨p,d,k,hk,hd,hp⟩ := h
refine ⟨.ex (p.rename bindSlots),d,k,hk,hd,?_⟩
intro N hN hdN e he
change (∃ x ∈ N, (p.rename bindSlots).Sat _ (cons x (mix e d))) ↔ _
apply exists_congr; intro x
apply and_congr_right; intro hx
rw [SentenceForm.sat_rename]
change p.Sat _ (cons x (mix e d) ∘ bindSlots) ↔ _
rw [bind_mix]
exact hp N hN hdN (cons x e) (by intro i; cases i; exact hx; exact he _)
theorem powerBound_domainDefinable (C : Context M) (a q : ℕ) :
DomainDefinable M (fun N e => PowerBound N (e a) (e q)) := by
refine ⟨FullSetForcing.ownPower (2*a) (2*q),fun _ => ZFSet.omega,1,
fun _ => ⟨0,by decide,rfl⟩,fun _ => C.omega_mem,?_⟩
intro N hN hd e he
rw [FullSetForcing.ownPower_spec N hN _ _ _ (by
intro i; unfold mix; split <;> first | exact he _ | exact hd 0)]
simp only [mix_even,PowerBound]
end TuringRigidity.BoundedDefinability
end OAI
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